Let $R$ be a polynomial ring with $n$ generators. The secondary invariants of a group $H$ in $GL(n)$ are the set of generators of the ring of invariants $R^H$ as a $K[\theta_1,\ldots,\theta_n]$-module. For this algorithm we always take the primary invariants $\theta_1,\ldots,\theta_n$ to be the elementary symmetric polynomials $e_1,\ldots e_n$ so that the ring $K[\theta_1,\ldots,\theta_n]$ is the ring of symmetric polynomials.
The secondary invariants are obtained by considering the quotient ring $R/(e_1,\ldots,e_n)$. This quotient ring is called the coinvariant algebre of $S_n$. This quotient is isomorphic to the regular representation of $S_n$. In particular as a K-vector space it is finite dimensional. In this space we find the subspace that is invariant under the action of $H$. The secondary invariants correspond to a basis for this space.
The advantage of this algorithm is that it decomposes the regular representation into its irreducible representation by means of the higher Specht polynomials basis. This reduces significantly de dimension of the vector spaces in which the invariant spaces must be found.
To illustrate we calculate the secondary invariants for a subgroup of cardinality 24 in $S_6$.
i1 : R = QQ[x_1..x_6] o1 = R o1 : PolynomialRing |
i2 : genList = {{1,2,3,0,5,4},{0,4,2,5,1,3}}
o2 = {{1, 2, 3, 0, 5, 4}, {0, 4, 2, 5, 1, 3}}
o2 : List
|
i3 : time seco = secondaryInvariants(genList,R);
(Partition{6}, Ambient_Dimension, 1, Rank, 1)
(Partition{5, 1}, Ambient_Dimension, 5, Rank, 0)
(Partition{4, 2}, Ambient_Dimension, 9, Rank, 1)
(Partition{4, 1, 1}, Ambient_Dimension, 10, Rank, 0)
(Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
(Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
(Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
(Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
(Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
(Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
(Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
-- used 1.52433 seconds
|
i4 : seco#(new Partition from {2,2,2})
2 2 2 4 2 2 2 2 2 2 2 2 4 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 4 2 2 2 2 2 2 2 2 4 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 1 2 2 2 2 2 1 2 2 2 2 2 4 2 2 2 2 2 2 2 2 4 2 2 2 2 2
o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x } }
3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6
2 3 2 2 4 2 3 2 2 2 2 3 4 3 2 2 2 2 3 2 2 3 2 2 2 3 2 2 2 2 3 2 4 2 3 2 2 2 2 3 4 2 2 3 2 2 2 3 1 3 2 2 2 2 3 2 1 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 3 2 2 2 2 3 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 1 3 2 2 1 3 2 2 2 3 2 2 1 3 2 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 1 3 2 2 2 2 3 2 1 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 1 3 2 2 1 2 3 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 3 2 2 2 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 2 3 4 2 2 3 2 2 2 3 2 2 2 3 4 2 2 3 2 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 3 2 2 2 2 3 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 1 3 2 2 1 3 2 2 2 3 2 2 1 3 2 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 2 2 3 2 1 3 2 2 1 2 3 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 3 2 2 2 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 3 2 2 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 4 2 2 3 2 2 2 3 2 2 2 3 4 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3
{0, 1, 2, 4, 3, 5} => HashTable{0 => -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x - -x x x x + -x x x x - -x x x x }
3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 4 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 3 5 3 1 2 3 5 3 1 2 3 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 4 5 3 1 2 4 5 3 1 3 4 5 3 2 3 4 5 3 1 3 4 5 3 2 3 4 5 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 3 6 3 1 2 3 6 3 1 2 3 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 4 6 3 1 2 4 6 3 1 3 4 6 3 2 3 4 6 3 1 3 4 6 3 2 3 4 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 2 5 6 3 1 2 5 6 3 1 3 5 6 3 2 3 5 6 3 1 3 5 6 3 2 3 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6 3 1 4 5 6 3 2 4 5 6 3 3 4 5 6
1 3 3 2 2 3 2 3 1 2 3 3 1 3 3 2 2 3 3 2 1 3 3 2 1 3 2 3 2 2 3 3 1 3 2 3 2 3 2 3 1 2 3 3 1 2 3 3 2 3 3 2 4 3 3 2 2 3 3 2 2 3 3 2 4 3 3 2 2 3 3 2 1 3 2 3 1 2 3 3 2 3 2 3 1 3 2 3 2 2 3 3 1 2 3 3 1 3 2 3 2 3 2 3 1 3 2 3 1 2 3 3 2 2 3 3 1 2 3 3 1 3 3 2 2 3 2 3 1 2 3 3 1 3 3 2 2 3 3 2 1 3 3 2 1 3 2 3 2 2 3 3 1 3 2 3 2 3 2 3 1 2 3 3 1 2 3 3 2 3 3 2 4 3 2 3 2 2 3 3 2 3 3 2 4 3 3 2 2 3 3 2 2 3 2 3 4 2 3 3 2 3 2 3 4 3 2 3 2 2 3 3 2 2 3 3 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 2 3 2 3 1 2 3 3 1 3 2 3 1 3 2 3 1 2 3 3 2 2 3 3 1 3 2 3 2 2 3 3 1 3 2 3 2 3 2 3 1 2 3 3 1 2 3 3 1 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 2 2 3 3 2 3 3 2 4 3 3 2 2 3 3 2 2 3 3 2 4 3 3 2 2 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 3 2 1 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 2 3 3 2 1 3 3 2 1 3 2 3 1 2 3 3 2 3 2 3 1 3 2 3 2 2 3 3 1 2 3 3 1 3 2 3 2 3 2 3 1 3 2 3 1 2 3 3 2 2 3 3 1 2 3 3 2 3 2 3 1 2 3 3 1 3 2 3 1 3 2 3 1 2 3 3 2 2 3 3 1 3 2 3 2 2 3 3 1 3 2 3 2 3 2 3 1 2 3 3 1 2 3 3 1 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 1 2 3 3 2 2 3 3 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 3 2 3 1 3 2 3 2 2 3 3 4 2 3 3 2 2 3 3 4 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 3 4 2 3 3 2 2 3 3 4 2 3 3 2 2 3 3
{0, 2, 1, 3, 4, 5} => HashTable{0 => - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x }
3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6
1 3 2 2 2 2 3 2 1 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 1 3 2 2 1 2 3 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 3 2 2 2 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 2 3 4 2 2 3 2 2 2 3 2 2 2 3 4 2 2 3 2 2 2 3 1 3 2 2 2 2 3 2 1 2 2 3 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 2 3 2 2 4 2 3 2 2 2 2 3 4 3 2 2 2 2 3 2 2 3 2 2 2 3 2 2 2 2 3 2 4 2 3 2 2 2 2 3 4 2 2 3 2 2 2 3 1 3 2 2 2 2 3 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 2 3 2 2 1 3 2 2 1 3 2 2 1 3 2 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 1 3 2 2 1 2 3 2 2 3 2 2 1 3 2 2 2 2 3 2 1 2 3 2 1 3 2 2 2 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 1 2 3 2 1 3 2 2 2 2 3 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 2 3 2 2 1 2 3 2 1 3 2 2 1 3 2 2 1 2 3 2 2 2 3 2 2 3 2 2 1 3 2 2 1 3 2 2 1 3 2 2 1 3 2 2 2 3 2 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 3 2 2 4 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 4 3 2 2 2 3 2 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2 2 3 4 2 2 3 2 2 2 3 2 2 2 3 4 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 1 2 2 3 1 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3 2 2 2 3 1 2 2 3
{0, 2, 1, 4, 3, 5} => HashTable{0 => -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x }
3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6
1 4 3 2 2 1 3 4 2 2 2 4 2 3 2 1 2 4 3 2 2 3 2 4 2 1 2 3 4 2 1 4 2 2 3 2 2 4 2 3 1 2 2 4 3 1 3 2 2 4 2 2 3 2 4 1 2 2 3 4 2 4 3 2 2 1 3 4 2 2 1 4 2 3 2 1 2 4 3 2 1 3 2 4 2 2 2 3 4 2 1 4 3 2 2 2 3 4 2 2 1 4 3 2 2 2 4 3 2 2 1 3 4 2 2 1 3 4 2 2 1 4 2 3 2 1 2 4 3 2 2 4 2 3 2 1 4 2 3 2 2 2 4 3 2 1 2 4 3 2 2 3 2 4 2 1 2 3 4 2 1 3 2 4 2 1 3 2 4 2 1 2 3 4 2 2 2 3 4 2 1 4 2 2 3 2 2 4 2 3 1 2 2 4 3 2 4 2 2 3 1 2 4 2 3 1 4 2 2 3 1 4 2 2 3 1 2 4 2 3 2 2 4 2 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 2 3 2 2 4 4 2 3 2 4 2 2 2 3 4 4 3 2 2 4 2 2 3 2 4 2 3 2 2 4 2 3 2 2 4 2 2 3 2 4 4 2 3 2 4 2 2 2 3 4 4 2 2 3 4 2 2 2 3 4 1 4 3 2 2 1 3 4 2 2 2 4 2 3 2 1 2 4 3 2 2 3 2 4 2 1 2 3 4 2 1 4 2 2 3 2 2 4 2 3 1 2 2 4 3 1 3 2 2 4 2 2 3 2 4 1 2 2 3 4 1 4 3 2 2 2 3 4 2 2 1 4 2 3 2 2 2 4 3 2 1 3 2 4 2 1 2 3 4 2 2 4 3 2 2 1 3 4 2 2 1 4 3 2 2 1 4 3 2 2 1 3 4 2 2 2 3 4 2 2 2 4 2 3 2 1 2 4 3 2 1 4 2 3 2 1 4 2 3 2 1 2 4 3 2 2 2 4 3 2 1 3 2 4 2 1 2 3 4 2 2 3 2 4 2 1 3 2 4 2 2 2 3 4 2 1 2 3 4 2 2 4 2 2 3 4 2 4 2 3 2 2 2 4 3 4 4 2 2 3 2 2 4 2 3 2 4 2 2 3 2 4 2 2 3 2 2 4 2 3 4 2 4 2 3 2 2 2 4 3 4 2 2 4 3 2 2 2 4 3 1 3 2 2 4 2 2 3 2 4 1 2 2 3 4 2 3 2 2 4 1 2 3 2 4 1 3 2 2 4 1 3 2 2 4 1 2 3 2 4 2 2 3 2 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 2 4 3 2 2 1 3 4 2 2 1 4 2 3 2 1 2 4 3 2 1 3 2 4 2 2 2 3 4 2 1 4 3 2 2 2 3 4 2 2 1 4 3 2 2 2 4 3 2 2 1 3 4 2 2 1 3 4 2 2 1 4 2 3 2 1 2 4 3 2 2 4 2 3 2 1 4 2 3 2 2 2 4 3 2 1 2 4 3 2 2 3 2 4 2 1 2 3 4 2 1 3 2 4 2 1 3 2 4 2 1 2 3 4 2 2 2 3 4 2 1 4 3 2 2 2 3 4 2 2 1 4 2 3 2 2 2 4 3 2 1 3 2 4 2 1 2 3 4 2 2 4 3 2 2 1 3 4 2 2 1 4 3 2 2 1 4 3 2 2 1 3 4 2 2 2 3 4 2 2 2 4 2 3 2 1 2 4 3 2 1 4 2 3 2 1 4 2 3 2 1 2 4 3 2 2 2 4 3 2 1 3 2 4 2 1 2 3 4 2 2 3 2 4 2 1 3 2 4 2 2 2 3 4 2 1 2 3 4 2 2 4 3 2 2 4 3 4 2 2 2 4 3 2 2 4 4 3 2 2 2 3 4 2 2 2 3 4 2 2 4 4 3 2 2 2 3 4 2 2 2 4 3 2 2 2 4 3 2 2 2 3 4 2 2 4 3 4 2 2 4 4 3 2 2 2 4 3 2 2 2 4 3 2 2 2 4 3 2 2 2 4 3 2 2 4 4 3 2 2 2 3 4 2 2 2 3 4 2 2 4 3 4 2 2 2 3 4 2 2 4 3 4 2 2 2 3 4 2 2 1 4 2 3 2 2 2 4 3 2 1 4 2 3 2 2 4 2 3 2 1 2 4 3 2 1 2 4 3 2 2 4 2 3 2 1 2 4 3 2 1 4 2 3 2 1 4 2 3 2 1 2 4 3 2 2 2 4 3 2 2 4 2 3 2 1 4 2 3 2 1 4 2 3 2 1 4 2 3 2 1 4 2 3 2 2 4 2 3 2 1 2 4 3 2 1 2 4 3 2 2 2 4 3 2 1 2 4 3 2 2 2 4 3 2 1 2 4 3 2 2 3 2 4 2 1 2 3 4 2 1 3 2 4 2 1 3 2 4 2 1 2 3 4 2 2 2 3 4 2 1 3 2 4 2 2 2 3 4 2 1 3 2 4 2 2 3 2 4 2 1 2 3 4 2 1 2 3 4 2 1 3 2 4 2 1 3 2 4 2 2 3 2 4 2 1 3 2 4 2 2 3 2 4 2 1 3 2 4 2 2 2 3 4 2 1 2 3 4 2 1 2 3 4 2 1 2 3 4 2 1 2 3 4 2 2 2 3 4 2 1 4 2 2 3 2 2 4 2 3 1 2 2 4 3 2 4 2 2 3 1 2 4 2 3 1 4 2 2 3 1 4 2 2 3 1 2 4 2 3 2 2 4 2 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 2 4 2 2 3 4 2 4 2 3 2 2 2 4 3 4 4 2 2 3 2 2 4 2 3 2 4 2 2 3 2 4 2 2 3 2 2 4 2 3 4 2 4 2 3 2 2 2 4 3 4 2 2 4 3 2 2 2 4 3 1 4 2 2 3 2 2 4 2 3 1 4 2 2 3 2 4 2 2 3 1 2 4 2 3 1 2 4 2 3 2 4 2 2 3 1 2 4 2 3 1 4 2 2 3 1 4 2 2 3 1 2 4 2 3 2 2 4 2 3 2 4 2 2 3 1 4 2 2 3 1 4 2 2 3 1 4 2 2 3 1 4 2 2 3 2 4 2 2 3 1 2 4 2 3 1 2 4 2 3 2 2 4 2 3 1 2 4 2 3 2 2 4 2 3 1 2 4 2 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 2 2 2 4 3 1 2 2 4 3 1 2 2 4 3 2 3 2 2 4 4 2 3 2 4 2 2 2 3 4 4 3 2 2 4 2 2 3 2 4 2 3 2 2 4 2 3 2 2 4 2 2 3 2 4 4 2 3 2 4 2 2 2 3 4 4 2 2 3 4 2 2 2 3 4 1 3 2 2 4 2 2 3 2 4 1 2 2 3 4 2 3 2 2 4 1 2 3 2 4 1 3 2 2 4 1 3 2 2 4 1 2 3 2 4 2 2 3 2 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 2 3 2 2 4 1 2 3 2 4 1 3 2 2 4 1 3 2 2 4 1 2 3 2 4 2 2 3 2 4 1 3 2 2 4 2 2 3 2 4 1 3 2 2 4 2 3 2 2 4 1 2 3 2 4 1 2 3 2 4 1 3 2 2 4 1 3 2 2 4 2 3 2 2 4 1 3 2 2 4 2 3 2 2 4 1 3 2 2 4 2 2 3 2 4 1 2 3 2 4 1 2 3 2 4 1 2 3 2 4 1 2 3 2 4 2 2 3 2 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 2 2 2 3 4 1 2 2 3 4 1 2 2 3 4
{0, 3, 1, 4, 2, 5} => HashTable{0 => - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x + -x x x x x + -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x + -x x x x x - -x x x x x + -x x x x x - -x x x x x - -x x x x x }
3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 5 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 4 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 3 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 2 4 5 6 3 1 2 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6 3 1 3 4 5 6 3 2 3 4 5 6
o4 : HashTable
|